Torque and Angular Momentum

Angular momentum is the rotational analogue of linear momentum. The central H3 idea is: ∑τₑₓₜ = dL/dt for components about a fixed axis. For a rigid body with constant I, this reduces to ∑τₑₓₜ = Iα. Here, “net torque” always means the resultant external torque about the stated axis.

1. Definitions (Must Know)

• Angular momentum, L:
  • particle about an origin: vector L = vector r × vector p
  • fixed-axis rigid body (magnitude): L = Iω
• Torque, τ about an axis/pivot:
  • magnitude: τ = rF sin φ = Fd_⊥
• Torque–angular momentum relation: ∑τₑₓₜ = dL/dt
• Symbols used in this lesson: vector r position vector (m), vector p momentum (kg m s⁻¹), vector L angular momentum (kg m² s⁻¹), τ torque (N m), I moment of inertia (kg m²), ω angular speed (rad s⁻¹), α angular acceleration (rad s⁻²).

Quick comparison:

2. Key Ideas (What Earns Marks)

• Always specify the reference point/axis when talking about torque or angular momentum.
• Torque changes angular momentum: no net external torque ⇒ angular momentum is conserved (about that axis).
• For fixed-axis rigid bodies: L = Iω is the high-speed exam tool.
• Variable I problems (ice skater): if τₑₓₜ ≈ 0, then Iω stays constant even though ω changes.

3. Detailed Explanations

A. Why the axis matters

Both τ and L are defined about a point/axis. Changing the axis changes:

• the moment arm d_⊥ (hence the torque), and
• the distance vector vector r (hence the angular momentum).

B. Linking torque and angular momentum (fixed-axis intuition)

For a point mass moving in a circle of radius r about an axis, the tangential force Fₜ changes the tangential speed, hence changes the “rotational motion”. The torque about the axis is: τ = rFₜ

For a rigid body about a fixed axis, all the parts rotate together with the same ω. The result you use is: ∑τₑₓₜ = dL/dt

If L = Iω and I is constant:

This connects directly to the previous two lessons:

• Dynamics of Angular Motion
• Moment of Inertia

C. When I changes (ice skater idea)

If there is negligible external torque about the spin axis, then dL/dt ≈ 0, so L is approximately constant.

For a spinning skater, you can model L ≈ Iω. Pulling arms in reduces I, so ω increases to keep Iω constant.

4. Common Mistakes

• Writing “angular momentum is conserved” without stating “when external torque is negligible (about the chosen axis)”.
• Using L = Iω for a particle not in rigid rotation about the axis (use vector L = vector r × vector p if needed).
• Forgetting that τ = Fd_⊥ depends on the perpendicular distance to the line of action (not just “the radius”).
• Mixing up L (angular momentum) with I (moment of inertia) because both appear in rotational formulas.

5. Exam Tips

• If the question screams “conservation”, check the external torque about the relevant axis:
  • Is there a pivot reaction through the axis? (often gives zero torque)
  • Is friction/air resistance negligible? (often assumed)
• Use the reasoning chain: negligible external torque ⇒ constant L ⇒ constant Iω for fixed-axis rotation.
• For fixed-axis questions, start from L = Iω and ∑τₑₓₜ = dL/dt; it’s faster than re-deriving.

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment: Vector Form and 3D Rotation

The full relation is vector-based: ∑ vector τₑₓₜ = (d vector L)/dt with vector L = vector r × vector p. For 3D rigid body rotation (changing axis direction), you need more advanced dynamics; H3 fixed-axis questions typically avoid this unless explicitly guided.

Next step

Return to the Rotational Motion hub, or continue in sequence to Rotational Kinetic Energy.